Dungeons & Dragons, if we talk about it as rules set, is ment for a party. Crew of adventurers, who all fill some purpose. In older editions this was more clear. One healed, one cleared traps, one fought, one was long range, one cast spells etc. Every character (class) had their function and place in the party.
For your average dungeon you most likely need them all. Lonely rogue cannot heal himself without stock of potions, warrior can easily run into traps and cleric can heal himself, but can he kill all the enemies?
Basically, all the characters in party are "one". Like human head. Ears to hear, nose to smell, mouth to taste, eyes to see. They are individual parts and alone not that effective, but as a "party", head they all function fulfilling eachother.
So, that quick idea about D&D party in mind, how does D&D work for solo play? I haven't played that much AD&D or older editions (I don't count few sessions or adventures) so I focus on D&D 3.X with what I have experience as DM (and DM character).
Party With One Player
You can make NPC's to fill the gaps in the party, or let your player to make several characters, as one as his/her "main" character to interact with. Other are like player controlled henchmen/NPC's. You might leave all or most of the interaction for this main or favourite character of player, or give player one character as others follow him as DM played characters. You might let player only control his own character in battles or let player to "move" all the characters in the party similar to tactical individual person war games (Necromunda for example).
If you as DM will control all NPC's and monsters in a combat encounter and let player only interact with his character, the problem is, that combat encounters might be a bit boring. First it's NPC 1, then NPC 2, then player character, then NPC 3, then NPC 4, then monster 1 and so on.
Depending on how many monsters in the encounter and NPC's there are in the party majority of the time player might just sit there and observe as DM rolls dice against himself. Boring, isn't it.
You can deal these situations like mini battle scenarios. Player vs. DM. Characters (as player) vs. Monsters (as DM). It is more interesting that way even if player doesn't have full control over NPC actions in non-combat encounters.
Similarly if there is a trap doors, locked crests etc. mysteries, do you really want to play against your own dungeon?
For example, player has fighter/warrior character, and this dungeon is full of traps and secret doors. You have rogue/thief NPC what handles those things. So basically you are playing against your own dungeon when player character just hangs around waiting for some skeletons to kill...
Easy way to skip this is just play social aspects of NPC's. Let player say, which NPC does or should do what. This way single player can cast spells, fight monsters, search for traps etc. It might be a big task for one player to handle with, so you might help him out with NPC's own thoughts.
If for some reason player with his fighter/warrior is not good at riddle solving, don't give easy xp. He has chosen to be fighter in your game, so he should do what he wanted to do - fight. If he does do other character classes things (find traps with NPC) reward for that. But if player just wants to kill and you as NPC's do all the brainwork, remember it in party xp.
Don't give xp too easily for single character from things he doesn't do but you as DM.
Multi-Classing Or Higher Level Start
CR... how hard particular monster is for a group of 4. If there truly is a single character, you might want to start him from higher level. It might still be challenging to kill lower level enemies without a party they are designed for. For lvl 4 character low CR skeletons can be tougher than for a party, so experience is well deserved.
If you start game with higher level character or character rises levels, he could concider multi classing. That way his character could become more all-around type with skills outside his original class.
If you don't want to make things a bit easier to single player, start with level 1. There is just this Computer-RPG-syndrome easily formed. Kill rats and other small critters, help lady to find his lost ring etc. That kind of game would be your average story from peasant to hero, but if you both like it, that's great.
You could discuss about what player is expecting from single player game. Does he want to play by rules, or does he want something more epic for his hero?
Good point in 3.X is that basically every character can learn almost any cross class skills, so they can be developed into all-around average adventurers. Only problem is, that if player shares his skills on everything, he's not necessarily good at anything.
More experience for one character is faster level development, sure, but do you make game harder as player levels up? If you keep up game's difficulty based on character level, without a party he might feel, that even though his character does advance in levels the game remains the same. You know new stuff, but so does enemies.
You might want to make it easier for player by giving magical or special items. For example my player has sorcerer character, but he also uses spear. Unfortunately, she wasn't that good in it and there were situations she would have liked to use it rather than magic.
So, did I tell that it's too bad, you wanted to be a magic user then be it and stop whining. If you wanted to be better with spear make a fighter? No.
I know that magic users are good at magic, and not that good in melee weapons as other classes. Still, my player wanted to mainly be a magician, not fighter. Still, when she wanted to hit enemies with her spear, he was quite frustrated as her character didn't handle that very well. Mainly she uses magic, yes, but sometimes when she wants to do something else than just cast spells it did not work.
So, I let her character to find +1 to spear attack ring. She was happy with it. +1 is not that big change actually, but still, it's something. She knows that her character is not a fighter, but even that +1 made her day (and the game).
So, give character special items. Don't give them too easy though, or every time your player is unsatisfied on some stat on his character sheet.
For a special quest you might want to give some items that quest giver wants back after quest, or some what only has temporary or limited effect. For fighter wand of healing with 5 charges after it withers to dust, for sorcerer potion of +5 attack for hard situation etc. Be creative.
Also ready to cast scrolls with one magic what can be used once can be okay to give. For opening locks etc.
But remember, don't stock your player's character with magical or special item for every situation and every stat. And don't give them too often or they are not special anymore. Player should think twice, before he uses his limited items, as they don't grow in the trees.
In single player game adventures which involve different classes and their individual skills could not work well with one player. If one player has full party to control, why not, but if he is single adventurer, it could be hard. So, design adventures for single player and for that particular character class.
Your player chose to be healing priest, don't make game monster mash.
Your player chose to be fierce barbarian, don't make game puzzle solving.
You can see from your player's character choise pretty well, what he would like to do in the game.
Make an interesting plot. For player party it might be easy just to put this random dungeon there with monsters for xp and random treasures and dragon in the end... that's easy, right? But for single character you might want to develope more plot and story than just character building.
If single player game is just grinding for levels and better items, it gets boring. Be creative, customize the game for the character class, make a real story.
Take some computer games, and see how they are done. Not necessarily computer RPG's but adventure and action games.
For example, if character is rogue, Thief games are great inspiration. Character can steal stuff, sneak, backstab, fight, solve mysteries while this huge plot is developing.
Of course for one shot games this is easier. Oh you are warrior, there are those orcs near the city. Thief, go steal inn keepers secret stash (when you find out where it is) and so on.
You can also make an investigation game. What kills cattle in night time? Why does the dead walk in the cemetary? Villagers are poisoned and sick, where is the source for this? Spy this merchant and find out what he is really up to. Again, with investigation game if player is fighter, you can put fighting in it and so on.
Even though D&D games are ment to played with party of characters from different classes, you can solo play it with one player. You can choose to use NPC's, give player more than one character to control or tailor adventure to be ment for single hero.
My D&D game with one player has contained NPC's to fill the party, but as player was newbie to this kind of roleplaying I controlled them (as explaining what is happening all the time so she learned during NPC actions). Some quests have been only solo tailored for character and I gave her character +1 to spear ring as she felt she sucks in it.
It's been fun. I bet it is more fun with player party and interaction between players in- and off-game, but still, you can have a full play of D&D with single player having fun!